From Pulset al. (2008)

USM USM

University Observatory Munich

XXIX Canary Islands Winter School of Astrophysics ‒

Applications of Radiative Transfer to stellar and planetary atmospheres

Radiative transfer in the (expanding) atmospheres of early type stars,

and related problems

Jo(achim) Puls

LMU Munich, University Observatory (USM)

USM

Content

Introduction (very brief)

Chap. 1 From plane-parallel to spherical atmospheres with velocity fields Chap. 2 RT: from p-p to spherical symmetry

Chap. 3 Line transfer in (rapidly) expanding atmospheres Chap. 4 Accelerated Lambda Iteration (ALI)

[and “pre-conditioning”]

Chap. 5 Further issues & applications (only keywords) Appendix A NLTE model atmosphere codes for hot stars Appendix B Further comments on the line-profile function

Specific text-books

Mihalas, D., “Stellar atmospheres (2

edition), 1975

Hubeny, I., & Mihalas, D., “Theory of stellar atmospheres”, 2014

XXIX Canary Island

USM

Why “expanding” atmospheres?

ƒ Observational findings:

early type star have outflows, at least quasi-stationary

ƒ only small variability of global quantities

ƒ have to be explained

ƒ diagnostic tools have to be developed

ƒ predictions have to be given

(M, v )

M, v , v(r)

‒ Morton & Underhill 1977

‒ Howarth (p.c.)

∆t several years

v_max≈2,500 km/s

Theory of

expanding

atmospheres

USM

driven by radiative line acceleration, supersonic outflows:

¾Radiative transfer in expanding media required, both to calculate line acceleration, and to

synthesize SEDs (quantitative spectroscopy)

7 5

10 ...10 Msun/ yr, v 200 ... 3,000 km/s

Mⁱ ≈ ^{− −} _∞ ≈

pioneering investigations by Lucy & Solomon, 1970

Castor, Abbott & Klein, 1975 (CAK) reviews by Kudritzki & Puls, 2000

Puls et al. 2008

dramatic impact on stellar evolution of massive stars (mass-loss rate vs. life time!)

line driven winds important for chemical evolution of (spiral) Galaxies, in particular for starbursts

transfer of momentum (=> might induce star formation), energy and nuclear-processed material to surrounding environment

Line-driven winds from early type stars

Prerequisites for radiative driving large number of photons and

large number of lines close to flux maximum required (typically some 10⁴...10⁵lines relevant)

… with high interaction probability

(=> mass-loss dependent on metal abundances)

XXIX Canary Island

USM

Two major issues …

… relevant for the radiative transfer in early type stars

ƒ sphericity

(affects radiation field and density)

ƒ velocity fields

(mostly affect line-transfer, due to Dopper-shift)

USM

Chapter 1 From plane-parallel to spherical atmospheres with velocity fields

light ray through atmosphere

lines of constant temperature and density (isocontours) curvature of atmosphere

insignificant for photons' path:

α = β

significant curvature:

α ≠ β

examples

solar photosphere / cromosphere solar corona

atmospheres of expanding envelopes (stellar winds)

“cool” main sequence stars of OBA stars, red giants and supergiants white dwarfs

as long as Δr / R << 1

=> plane-parallel (p-p) symmetry

XXIX Canary Island

spherical symmetry

USM

e_z, e_r

e , e

e_y, e_Φ P at

r P

n

Co-ordinate systems/symmetries

Cartesian spherical

, , ,

specific inte

,

( , , , , , ) nsity

right-handed, orthonor

( , , , , :

important

m l

, ) a

x y z r

I x y z ν t I r ν t

Θ Φ

Θ Φ

= + + = Θ + Φ +

Θ Φ

x y z r

x y z r

r e e e r e e e

e e e e e e

n n

physical quantities depend plane-parallel

.... depen spherically sy

d only on , e.g.

symmetrie

mmetr

only s

ic

on

z , e.g.

( , , , ) ( , , , ) ( , , , ) ( , , , )

I t I z t I t I r

r

ν → ν ν → ν t

r n n r n n

intensity has direction into

requires additional angles with respect to , , , , and

= ( , )

,

r

d θ φ

θ

Θ Φ

Ω

x y z

z

n

e e e e e e

e n n

= ( , ) ( , , , , , )

p-p symmetry

in

( , , , , , )

spherical sy d

mmetry

r

I x y z_ν t I_ν r t

θ

θ φ Θ Φ θ φ

e n

( , , )

ependent of az

imuthal dire

ct

ion

, ( , , )

I z_ν t I r_ν t

φ

θ θ

→ →

USM

Hydrostatic equilibrium

In (e.g., Kurucz atmospheres), assumed;

also in atmospheric models of ea

hyd rly

rostat type

ic equilibri stars with

plane-parallel atmospheres without wind

very thin winds e.g., TLUSTY (Hubeny

s um

[

]

(

)

g

*

grav 2 *

ra

g a

v

*

as r d

,

8) or DETAIL/SURFACE (Butler & Giddings 1985), see Appendix A

with , assuming z(photosphere) R

Integration gives either wit

( )

h (

(

a

) )

tot

tot

G

p z g g M

g R

P P

z z

P z g

P

m

∂ =ρ − = Δ

=

⋅ +

∂ +

=

nd mass column density ( )d

z

m=^∞

∫

rad

B sound

H

2 sound * B *

2 *

H grav es

/

c

2v ( ) or, neglecting , with photospheric scale height

v

v isother

= is the [order of few km/s], the mean mol

( ) (

mal speed of so ecular w

0

u ) e

d ,

n g z H

k T

z z k T

H R

g m

T ρ ρ m

μ μ

μ

≈ = − = =

* esc

*

rad

eight, and

v 2 the [usually, order of several 100 km/s]

Alternatively (again neglecting

( ) 1 log log log( )

photospheric escape velocit ),

, i.e.,

y

m m m H

H GM

R

g

ρ ≈ ρ = −

=

XXIX Canary Island

USM

[ ] [ ]

stationarit , i.e., 0 an

y

(

d ext

)

inclusion of velocity fields Equation of continuity:

Equation of momentum ("Euler equati

Hydrodynamic desc

o

ription:

( ) 0

"

) n ) (

t

t

t p

ρ ρ

ρ ρ

ρ ρ ρ

∂=

+

∂

∇⋅ ⋅∇

∂ + ∇ ⋅ =

∂

∂ + ∇

⇒

⋅ ⊗ = −∇ +

∂ v v v v

v

v v v g

2

2 "advection term",

(from ine 2

spherical symm ext

r ,

t 1 etr

i.e., (

y

ia) )

v const

4 (I) with (

v (II)

0 v

)

r ur

r r

r

r M

p g

r r

ρ

ρ π

ρ ρ

∇⋅ → ∂

∂

= =

∂ = −∂

∇ ⋅ =

∂ ∂ +

u

v

Hydrodynamic description

* 2 Rad

( v

I: Conservation of mass-flux II: "Equation of motion"

with gravity and radiative acceleration

or

, to be compa

)v( ) ( ) (

re w

) d

p GM

r r r g r

r r r

ρ ^{∂ ∂} ρ ^{⎛ ⎞}

⇒ ∂ = −∂ + ⎜⎝− + ⎟⎠

* 2 Rad

* 2 Rad

*

( )v( ) v

ith hydrostatic equilibrium

hydrostatic equilibrium in p-p symmetry

( ) ( )

( ) ( )

: GM

p r g r

r r

p GM

z g z

z R

r r ρ r

ρ ρ

⎫⎬

⎭

∂ = ⎛⎜− + ⎞ ∂

∂ ⎝ ⎟⎠

⎛ ⎞

∂∂ = ⎜⎝− + ⎟

∂

⎠

−

USM

When is (quasi-)hydrostatic approach justified?

(

)

2 2

B

sound H

nd 2

By using (equation of state) , and (for the hydrodynamic case)

the equations of motion and of hydrostatic equilibrium can be rewritten:

v

= v 4 v const

( v ( )

p k T M r

m

r r

r ρ ρ

ρ ρ π ρ

= μ =

∂ = −

∂

=

− _{grav Rad} ^2sound

2

2 sound

sound grav * d

2

Ra

dv [hydrodynamic]

[hydrostatic, p-

) ( ) ( ) 2v

d

v ( ) ( ) ( ) dv p]

dz

( )

g r g r

r

z g R z

r z g

r

ρ ρ

⎛ ⎞

− +

⎜ ⎟

⎝ ⎠

⎛ ⎞

∂∂ = − ⎜⎝ − + ⎟⎠

−

Conclusion:

‰ for v << v

, hydrodynamic density stratification becomes (“quasi”-) hydrostatic

‰ this is reached in deeper photospheric layers, well below the sonic point, defined by v(r

)=v

Thus: p-p atmospheres using hydrostatic equilibrium give reasonable results

even in the presence of winds as long as investigated features (continua, lines) are formed below the sonic point (see also slide 12)

XXIX Canary Island

USM

Unified atmospheres ‒ density/velocity stratification

for stars with winds

photosphere + wind = unified atmosphere (Gabler et al. 1989)

Two possibilities:

a) stratification from theoretical wind models [Castor et al. 1975, Pauldrach et al. 1986, WM-Basic (Pauldrach et al. 2001), Appendix A]

Disadvantage: difficult to manipulate if theory not applicable or too simplified b) combine quasi-hydrostatic photosphere and empirical wind structure [PHOENIX

(Hauschildt 1992), CMFGEN (Hillier & Miller 1998), PoWR (Gräfener et al. 2002), FASTWIND (Puls et al. 2005), Appendix A]

Disadvantage: transition regime ill-defined

grav rad

sound sound 2

*

at first (r) calculated (quasi-hydrostatic, with g ( ) and g (r)) v( ) for v v (roughly: v < 0.1

deep layers:

outer l

v )

4 ( )

at first v(

ayers: r) = v (1 )

r r M

r r bR

r

β

ρ

π ρ

∞

→ =

−

2

, "beta-velocity-law", from observations/theory (b from transition velocity) ( )

4 v( )

: smooth transition from deeper to outer stratification Input/fit param

transition z

eters:

one r M

r r

ρ π

→ =

, M v , , _∞ β location of transition zone

USM

Unified atmospheres ‒ density/velocity stratification

XXIX Canary Island

dotted: hydrostatic

solid: unified model with thin wind

dashed: unified model with dense wind dotted: hydrodynamic, from wind-theory solid: unified model, with similar v_∞ and β=0.8 corresponds to log ρ = log m – log H

cores of strong lines

From Puls et al. (2008)

NOTE: at same τ or m, wind-density (for v ≥ v

) lower than if in hydrostatic equilibrium

USM

Plane-parallel or unified model atmospheres?

‰ Unified models required if τ

≥ 10

at transition between photosphere and wind (roughly at 0.1*v

)

‰ rule of thumb using a typical velocity law (β=1)

‰ if

→ plane-parallel, hydrostatic models possible for optical spectroscopy of late O-dwarfs and B-stars up to luminosity classes II (early subtypes) or Ib (mid/late subtypes)

‰ check required!

2

max Ross sound

8 1

1

( 10 at 0.1 v ) v

10 100

6 10 R 0

M M

R kms

τ

⋅

M yr

= = ≈ ⋅ ⋅

(actual) <

for considered object,

then (most) diagnostic features formed in quasi-hydrostatic part of atmosphere

M M

USM

Chapter 2

RT: from p-p to spherical symmetry

XXIX Canary Island

specific intensity and moments similarly defined if z→r

1 th

1

( , ) ( , ) with =cos and = ( , ) [in the following, - and -dependence suppressed]

from symmetry about azimuthal direction:

n moment = 1 ( , ) d , a s in p-p case when ; n=0,1,2 2

n

I

z

z I

r

r t

I r J

μ μ μ θ θ ν

μ μ μ

+

−

→

→

∫ →

e n

( ), ( ), ( )

: only r-component different from zero, pro flux(-density)

radiation stress te

pto Eddington-flux : only diagonal elements different from ze

nsor o

0 0

r

=

4 H

r H r K r

π

⎛ ⎞

⎜ ⎟

⎜ ⎟

⎜ ⎟

⎝ ⎠

P F

( )

refers to

, only z-component different from zero, and in pp-symmetry

in spherical s

wit 4 only difference

h (radia

divergence tion pressu

of radiation stress tensor,

z re scalar) = (

y e

) mm tr

z

p K z

c

p π

∇ ⋅ = ∂

∂

∇ ⋅

P

P

( )

y, only r-component different from zero, and with (r 4

3 adiation energy density) = ( )

r

p p u

r u

r J r

c π

∂ −

∇ ⋅ = +

P ∂

USM

( )

*

, phot

*

+1 +1

*

1 th *

* t

1

envelope optically thin const

radiation field leaving photosphere isotropic: ( ) const ( )

1 1

( ) d ( ) d

2 assu

1 1

n moment (

2 e.g.,

2 ) 1

( me:

0

n

n n

I

I I R

I I R I R

n J

ν

ν

ν ν

ν

ν

ν μ

μ μ

μ μ μ μ μ

+ +

+

+

−

+

→ =

= =

⇒ −

=

= →

∫ ∫

h

2

* *

*

*

* *

with "dilution factor,

1(1 ) and 1 (cone angle subtended by stellar disk, s

moment )

in )

2 )

( W

WI R

R R

W r r

ν

μ μ θ

+

⎛ ⎞

= − = −⎜⎝ ⎟⎠ =

≈

optically thin envelopes

2 n+1

*

2

*

*

*

*

*

Now, 1 1 , and moment

2

in other words, all Eddingf

This is specific for

actors (

for , any

.... 1 ( )

r

(sp

atios of moments) 1 fo

herical) e 4

r

nve r R

J H K I R R

r

r r

R R

n

ν ν ν ν

μ

+

⎛ + ⎛ ⎞ ⎞

⎛ ⎞

= = = → ⎜⎝ ⎟⎠

→ −⎜⎜⎝ ⎜ ⎟⎠ ⎟⎟⎠

→

⎝

lopes at large distances from the star, and different from corresponding plane-parallel results

USM

Equation of radiative transfer

XXIX Canary Island

general case:

1 d

( , , ) ( , , ) ( , , ) ( , , ), with directional deri

plane-parallel, stationar

vative

c d

d (actual path is longer than height differen :

d

y

ce,

I t t t I t

t s

z ds

ν

η

χ

μ

⎛ ∂ + ⋅∇ ⎞ = − ⋅∇

⎜ ∂ ⎟

⎝ ⎠

⋅∇ ⇒ =

n r n r n r n r n n

n

d ( , ) ( , ) ( , ) ( , )

/ ) :

d I z z

dz

z I z

z

μ μ η χ μ

μ

μ μ

= −

μ'≠ μ

2

2

(1 )

(can be shown by spherically symmetric,

using a "p-z geometry"

stationary ( no longer constant along direction

(1 )

, se ( , ) ( , ) ( , ) (

e belo ,

:

)

)

w)

I r r r I r

r r

r r

ν ν ν ν

μ μ μ η μ χ μ

μ μ

μ μ

μ μ

⎛ ⎞

∂ − ∂

⋅∇ ⇒

∂ − ∂

+ = −

⎜ ∂ ∂ ⎠ ⎟

∂ ∂

⎝

+

n

n

USM

Moments of the RTE

( )

general case, th

plane-parallel, stationary and static

spherically symmetric, stationary and (quasi-)static [no/negligible Dopplershifts no

4 d

winds 0 mome

or c c

d d

nt

onti

J I

t

H J

z

ν ν ν ν ν

ν ν ν ν

π η χ

η χ

⇒

∂ + ∇ ⋅ = − Ω

∂

= −

∫

F

2 2

nuum problems(except for edges)

Otherwise, opacities become angle-dependent (Doppler-shifts), and cannot be put in front of the integral

1 )

s]

(r H J

r r

ν ην χν ν

∂ = −

∂

( )

t

2

1 momens

1 1

c d

general cas

d d

3

e, t

t c I

K H

z

K K J

r r H

ν ν ν ν

ν ν ν

ν ν ν

ν ν

η χ

χ

χ

∂ + ∇ ⋅ = − Ω

∂

= −

∂ −

+ = −

∂

∫

P n

F

when frequency integrated, = 0, if ONLY radiation energy transported:

radiative equilibrium, flux conservation

when frequency integrated, = -f

USM

r₄

4_i 0 μ =

41 1

μ =

4 -1 4 -1

4 i i

z μ = r

ray-by-ray solution ‒ p-z geometry

XXIX Canary Island

the following method ( ) works

ONLY for spherically symmetric problems and no Doppler-shifts!

a) define p-rays (impact-parameter) ta

based on Hummer & Rybicki 1 ngential to each discrete r

971 adi NOTE:

al s

2 2

hell b) augment those by a bunch of (equidistant) p-rays resolving the core c) use only the forward hemisphere, i.e.,

and 0

all points , 1, NP, are located on the same -shell,

di d i di

di d

z r p z

z i r

= − >

⇒ = i.e., have the same

physical parameters such as emissivities, opacities, velocities, ...

(due to spherical symmetry, and neglect of Doppler-shifts) Now one solves the RTE : from first principles,

d ( , )

( ) ( ) ( , ) (with ' ' for 0 and ' ' for 0) d

using appropriate boundary conditions (core vs. non-core rays), a

along each p-ray

n

i

i

I z p

r r I z p

z

ν^± ην χν ν^± μ μ

± = − + > − <

d standard methods (finite differences etc.)

di di

After being calculated, ( ( ), ), 1, NP, samples the specific intensity at the same radius, , but at different angles, , starting at 1 for 1 and 1, NZ (central ray, =0)

di d i d

di

i d

I z r p i r

z i d p

r

ν

μ μ

± =

± = = = = to _di 0 (tangent ray, where and thus 0).

In other words, along individual -shells, the specific intensities ( , ) ( , ) are sampled for all relevant , and corresponding moment

i d di

d d d

p r z

r I r_ν I z_ν

μ

μ μ μ

± ±

= = =

= s can be calculated by integration.

USM

Feautrier-variables

+ -

In fact, the RTE is not solved for seperately, but for a linear combination of and , using the so-called Feautrier-variables and v , which allows to construct a 2nd order scheme as in the

I I I

u

ν ν ν

ν ν

±

+ -

+ -

plane-parallel case: higher accuracy, diffusion limit can be easily represented ( , ) 1( ( , ) ( , )) mean intensity like

2

v ( , ) 1( ( , ) ( , )) flux like 2

v ( ),

u z p I z p I z p z p I z p I z p

S u

z

ν ν ν

ν ν ν

ν χν ν ν

= +

= −

⇒ ∂ = −

∂

2 2

v

(2nd order, with ) u

dz

u^ν u_ν S d dz

ν ν ν

ν ν

ν

τν

χ

τ χ

∂ = −

⇒ ∂ = −

= −

∂

... and corresponding boundary conditions

inner boundary: for core rays, first order, using the diffusion approximation; for non-core rays, 2nd order, using symmetry arguments outer boundary: either I_ν^-( _max, ) 0, (e.g., shortward of HeII Lyman edge)

Formal solution for ( ) (or ( ) and v

or higher order ( ) ) and co

for optically thic

rresponding angle-averaged quantities ( k conditi

mo o

me t ns

n s)

z p

I_ν μ u_ν μ _ν μ

=

affected by inaccuracies, due to specific way of discretization, but ratios of moments much more precise (errors cancel to a large part)

USM

Continuum transfer in extended atmospheres

XXIX Canary Island Thus:

solve the moments equations (only radius-dependent), and use Eddington-factors from formal solution to close the relations. Ensures

variable Eddingt

high accuracy ( on-factor

since dire method

ct solution for angle-averaged quantities, and 2nd order scheme), whilst Eddington-factors (from the formal solution) quickly stablilize in the course of global iterations.

Additional advantage: when using moments equations, optimum diagonal accelerated lambda-operator (see Chap. 4) can be easily calculated.

th st

2

2

Using the 0 and 1 moment of the RTE (see ) and / , we obtain

( )

( )

( ) (3 1)

f K J

r H r J S

f J f J

r H

ν ν ν

ν ν ν

ν

ν ν ν ν

ν

ν ν

τ

τ χ

=

∂ = −

∂

∂ −

− =

∂

slide 16

[ ]

core

2 2

core

2

2

Introducing a "sphericality factor" via ln( ) (3 1) /( ' ) d ' ln( ), the 2nd equation becomes

( )

, and can be combined with the first one to yield a 2nd order

r

r

q r q f r f r r

f q r J

q r H

ν ν ν ν

ν ν ν

ν ν

τν

= − +

∂ =

∂

∫

2

2 2

2 2

2 2

2 2 *

scheme for

with d d [for comp.: , l

( ) 1

( ) in p-p ( )

( ), imit for 1 and ]

X q f J q

r J

f q r J

r J S

X J r

q S R

ν

ν ν ν

ν ν ν ν

ν ν

ν ν

ν ν

ν

ν τ ν

τ

= → →

∂ =

− ∂

∂

∂ −

=

USM

Problem (to be detailed below)

When standard (observers’s frame) RT-methods applied, very high resolution in radial grid (∆v=O(v

/3)) required

(for specific methods, also very high resolution in μ required).

E.g., for v

=2000 km/s, and v

(O)=8km/s, N=750 radial grid points!

(problem becomes mitigated, when large “micro-turbulence” of order 100 km/s ‒ due to inhomogeneous wind structure ‒ considered)

NOTE as well:

Use only RTE (maybe cast into “Rybicki form” if separation into scattering and thermal part possible), but do NOT use moments equations as before, since only general

formulation (top of slide 16) valid if opacities strongly μ-dependent (due to Doppler-shifts) In the following, we mostly consider the pure line case (except when stated differently), assuming that the continuum is optically thin (not so wrong for “normal” OB-star winds, but invalid, e.g., for WR-stars with much larger mass-loss rates).

Moreover, we assume pure Doppler-broadening, which captures the essential contribution when calculating occupation numbers etc. (→ scattering integral J̅̅ ).

For the calculation of emergent profiles, other broadening functions might/should be used if necessary (e.g., Stark- and Voigt-profiles)

Chapter 3

Line transfer in (rapidly)

expanding atmospheres

USM

Notation:

line-opacities/profile function

XXIX Canary Island

2

th

L D

D D 2

L lu th

e

v ( )

( ) ( ) ( , ) with ( , ) 1 exp and ( ) for a Doppler profile,

( ) ( )

( ) = ( _{l u} ^l ), is the line-center frequency, and v ( ) i

u

r r r r r r

r c

r g

r e f n n r

m c g

ν

ν

χ χ φ ν φ ν ν ν ν

ν π ν

χ π ν

⎡ ⎛ − ⎞ ⎤

⎢ ⎥

= = Δ ⎢⎣−⎜⎝Δ ⎟⎠ ⎥⎦ Δ =

−

[ ]

[ ]

ncludes any micro-turbulence if presen and no

t.

NOTE: cm s χ = tcm , φ =s

Due to the apparent Doppler-shifts (material is moving w.r.t. stellar rest-frame), absorbing/emitting ions "see"

the radiation field at corresponding comoving frame (CMF) frequencies, and absorb/emit

CMF

CMF

photons according to ( ), when the observer's frame frequency is , and ( ) v( ) in spherical symmetry.

Thus, the profile function has to be evaluated at the CMF-frequency,

( ,

c r

r

ν ν ν ν μ

φ ν

≈ − n v r⋅ n v r⋅ =

2

D D

1 v( ) /

) exp .

( ) ( )

r c

r r

ν ν μ ν ν π ν

⎡ ⎛ − − ⎞ ⎤

⎢ ⎥

= Δ ⎢⎣−⎜⎝ Δ ⎟⎠ ⎥⎦

th th

For simplicity, in the following we assume a spatially constant thermal speed, v , and measure frequencies in Doppler-units w.r.t. to this speed (a generalization to depth-dependent v (r) is provid

th D

D

D th th

ed in );

with v .

v( ) / v( ) v

Then, = v'( ) with v'( ) (0, 1].

Appendix

v

B

v

.1

x c

r c r

x r r

ν ν ν ν

ν

ν ν μ ν μ

ν ^∞

= − Δ =

Δ

− − − = ∈ >>

Δ

USM

Notation:

line-opacities/profile function

( )

[ ]

2 CMF

D

In this notation,

( , ) ( v', ) 1 exp v'( ) ,

the profile function has still units "per frequency", =s, and is only expressed with argument .

x r x r x r

x

ν ν

φ φ μ μ

ν π

φ

⎡ ⎤

= − = Δ ⎣− − ⎦

1

To simplify the following considerations, we include the factor ( D) from above into the opacity;

then the profile function has units "per Doppler-shift" (i.e., it's dimensionless and normalized w ν ⁻

Δ

( )

-1

L D

L 2 L L

CMF CMF CMF

D D th

th th

.r.t. ), whilst [ ( ) / ] [cm ]

( ) 1 ( ) ( )

( , ) ( , ) with ( , ) exp v'( ) , and ;

v

v v

Note that since v'( ) [ , ], needs to vary in the sam

v v

x r

r r r

x r x r x r x r

r x

ν

χ ν

χ χ χ λ

χ φ φ μ

ν π ν

μ ^{∞ ∞}

Δ =

⎡ ⎤

= Δ = ⎣− − ⎦ Δ =

∈ − + e range essentially,

(

)

over a few thermal Doppler widths.

⎡ x∈ −∞ ∞ ⎤

⎣ ⎦

For various integrals involving Φ, see Appendix B.2

USM

The resonance-zone

XXIX Canary Island

line processes only effective in a very localized region, the so-called

resonance-zone, whenever Φ(x_CMF) is non-negligible, i.e. when

(x-μv‘) ϵ [-∆x_Dop,+∆x_Dop] ≈ [-3,3]

d v'( ) d

Since v'( ) part of argument of , we need to know along path d (remember: )

d d

To calulate this quantity, we again make use of the p-z geometry in spherical symmetry (rotate such t

r r s

s s

μ φ μ n⋅∇ =

2 2

hat )

d v'( ) d v'( ) dv' d d dv' v' dv'

v' (1 ) 0 for v' > 0 and 0!

d d d d d d d

[In contrast to , 0 implies here 0, i.e., for negative angles we consider the back-hemisp

p p p

z

r r r

s z r z z r r r

z

μ μ μ μ μ μ

μ

→ = + = + − > >

< <

n

In spherical symmetry, v'( ) increases monotonically along any given direction , as long as v' 0 is monotonically inc here]

reasing.

μ r n >

slide 17

I(x

+Δx)

[μv‘](z)= μ(z)v‘(r(z)), i.e.,v‘>0 always (for outflows) I(x

)

I(x

)

I

exp[-τ

(z

)] for pure abs.

I(x

-Δx)

of resonance zone for x₁=μv‘₁ (blue edge) begin center end (red edge)

I

μv‘₁(z₁) μv‘₁-Δx μv‘₁+Δx

1 1 1 1 varie

( ) ( , ) ( v' , ) d s

I z =

∫

( [ ] )

min

z L

D

( ) ( ') v' ( '), ' d ' varies

x z

z z

z χ x z z z

τ φ μ

= ν −

∫

USM

The resonance-zone

Both freq. grid (x) and μv‘(z) need to be highly resolved, on scales corresponding to v_Dop.

‰ if μv‘(z)-spacing too coarse: resonance zone missed or not resolved, intensity remains constant (or too large), and I̅ becomes too large! (red, blue and green curves would become constant at I₀)

‰ if x-spacing too coarse: variation of I(x) (“neighboring” resonance zones) not sufficiently sampled.

(blue/green curves might be absent, and red curve not centered, if no frequency where x-μv‘=0)

‰ In spherical geometry, the first point is a specific problem, since the general spacing has to be provided for the radial grid (and not for specific p-rays), and a high resolution in v’(r) does not guarantee a high resolution in μv‘(z).

‰ In models using Cartesian co-ordinates (μ=const along a specific ray), this leads to the condition that ∆μ=∆x/v‘_max (intricate coupling of frequency and angle!)

I(x

+Δx)

[μv‘](z)= μ(z)v‘(r(z)), i.e.,v‘>0 always (for outflows) I(x

)

I(x

)

I(x

-Δx)

of resonance zone for x₁=μv‘₁ (blue edge) begin center end (red edge)

I

μv‘₁(z₁) μv‘ -Δx μv‘ +Δx

1 1 1 1 varie

( ) ( , ) ( v' , ) d s

I z =

∫

( [ ] )

min

z L

D

( ) ( ') v' ( '), ' d ' varies

x z

z z

z χ x z z z

τ φ μ

= ν −

∫

I

exp[-τ

(z

)] for pure abs.

USM

Sobolev theory

XXIX Canary Island

From the slides before, it is evident that line processes (contrasted to continuum ones) occur in a very localized region in the wind. V. Sobolev (1960; but work done during world- war II) was the first to obtain a completely local approximation which is quite close to reality (and can be extended to become even more precise).

The following derivation follows (in part) Owocki & Puls (1996);

for an alternative derivation (very insightful), see Rybicki & Hummer (1978)

th th

l

For simplicity, in the following

(i) we concentrate on outflows, i.e., v( ) 0 [though dv/d 0 is not excluded], (ii) adopt, as before, a spatially constant thermal speed v ( ) v , and

(iii) define (

r r

r χ

> <

=

L D

) ( )r

r χ

= ν Δ

( )

1 2

1 2

1 2

max( , )

2 2

1 2 l

min( , )

The optical depth difference (along impact parameter ) between two points and is given by ( , , , ,) ( ') 'v'(r ') d ' with (as usual) ' ', and ' ' .

'

z z

z z

p z z

t x p z z r x z z r z p

χ φ μ μ r

=

∫

*

( , , , )

( , , , ) - ( , , , ')

core 0

direct component, only present diffuse component (from radiation scattered or em for >0 and

Then, without any approx.,

( , , ) e ^{t x p z zB t x p z zB} ( ')e^{t x p z z} d ( , , , ')

p R

I x p z_ν I S r t x p z z

μ

−

≤

= +

∫

itted in the wind)

for 0, with

else

The above equation is valid for both outward ( 0) and inward ( 0) directed rays, in dependence of the sign of . Here, we us

B

z z p R

z

μ μ z

> ≤

= ⎨⎧⎩−∞

≥ <

e a p-z geometry extending over both hemispheres, with 0 for the front z > and z<0 for the back hemisphere!

USM

Sobolev theory

*

( , , , )

( , , , ) - ( , , , ')

core 0

direct component, only present diffuse component (from radiation scattered or emitted in the wind) for >0 and

( , , ) e

( ')e

d ( , , , ')

p R

I x p z

I S r t x p z z

μ

−

≤

= + ∫

( )

* *

1

1

for 0, with

else To calculate the scattering integrals, we first integrate over ( )d

( , ) ( , , ) v'( ) d and then over d , ( ) 1 ( , )d

2

B

z z p R

z

x x

I r I x r x r x

J r I r

ν

φ

μ μ φ μ μ

μ μ

+∞

−∞

+

−

> ≤

= ⎨ ⎧ ⎩ −∞

= −

=

∫

∫

( )

the integrands provide a contribution only if ' v'( ') v'(r), respectiv

Now we consider that or

due to the behaviour of . For the optical depth difference, ( , , , ,) ( ') ' v'(r ')

ely

' d

,

B l

t x p z z r

x

x z

x r

φ

μ

μ χ

μ φ

= −

≈ ≈

( )

[ ]

max

min

z max( , )

l 0

min( , ) z

0

2

0 2 0

0

( ) ' v'(r ') d ',

where is the position of the corresponding resonance zone, and needs to be calculated from v' ( ) , i.e., 1- v'( ) (non-linear eq

B

B

z z

z z

r x z

r

r x p r x

r

χ φ μ

μ

→ −

= ± =

∫ ∫

.),

which has a unique solution for strictly monotonic flows (otherwise there is more than one resonance zone).

This is the Sobolev approximation:

opacities (and source functions) are assumed to be constant over the

resonance zone

USM

Sobolev theory

XXIX Canary Island

CMF

2 2

CMF

Moreover, we switch from an integration over d ' to an integration over CMF-frequency,

d d( v'(r'))

d d( v') dv' v'

(see ) (1 ) : ( , ). For 0 ,

d d d

by consi

p p

z

x x

x Q r

z z r Q

r μ

μ μ μ μ

= −

⎛ ⎞

= − = = − ⎜ ⎝ + − ⎟ ⎠ = − >

CMF CMF

dering boundaries: ( ) v'(r), ( ) [blueward of blue edge of resonance zone], and by putting ( , ) in front of the integral [same argument as before], we arrive at

x z x x z

Q r

μ μ

= − → ∞

slide 23

0 0,

0 L 0

0 0

2 2

0 0

D

This result can be generalized for , if we define

In the general case, is the directional derivative

( ) ( )

( , )

positive and negative values

of th

dv' v'

( , ) (

d

of

1

Q

)

r l

S

r r Q

r Q r

r r

χ χ

τ μ

μ ν μ μ

= =

Δ + −

l

e velocity in direction ,

i.e., ( ') = dv' if has direction Sobolev optical depth

d as the , evaluated at the resonance zone.

Q l

l

⎧ ⎪

⎪⎪ ⎨

⎪ ⎪ = ⋅∇ ⋅

⎪⎩

n

n n v n

CMF

CMF B

(z)

l 0

l 0 CMF CMF

0 0

(z ) v'(z)

S 0 0

( , , , ) 1 ( ) ( , ) ( v'(

( ) ( )d ( )d =

( ', ') ( ))

, ) r

x

x x

B

r x x r

Q r Q

t x p

z z r r x

μ

χ φ χ φ ξ ξ τ

μ μ

μ μ

−

− Φ −

≈ ∫ → ∫

USM

Sobolev theory ‒ notes for advanced reading

core

(rememb ( )

er

0 [ : v

blue(s 0)

NOTE1: , thus:

tarward) side of resonance zone]

( , , , ) 0 for "b

( ) 1 [red side of resonance zone]

efore" resonance zone,

( ,

) (

t x p z z

z I t I

t x p

Φ − Φ

→

∞

=

=

=

>

∞

core S

th D

NOTE2: for pure Doppler-profiles, ( ) 1 erfc( ) 2

NOTE3: since v' is the v

, , ) for "behind" res

velocity in units of t

onance zone, ( )

he thermal spee

s

d

e

,

ex

a

e

n d since

p( )

,

B S

z z z I

x x

t I

τ τ

ν λ

⎫ ⎬

⎭

Φ =

Δ =

→ ≈ −

[ ]

0, 0

L 0 -1

0 0

2 2

we can alternatively write

( , ) ( ) when v and are measured in actual units (then, v/ s )

dv v

(1 )

d

S

r r r r

r r

χ λ

τ μ

μ μ

= =

+ −

slides 23 / 24

USM

Sobolev theory

XXIX Canary Island

( )

( , , , ) - ( , , , ') ' ( , , , )

0 0

0 0

Since also the integrand of the diffuse component contributes only for 'v', ( ')e d ( , , , ') ( ) e d ' = ( ) 1 e ,

the specific intensity ca

B B

t x p z z t x p z z t t t x p z z

x

S r t x p z z S r t S r

μ

−

−

≈

→ −

∫ ∫

( )

S( ,0 0) CMF S 0 0 CMF

c

( ) ( , ) (

ore 0

)

n be approximated by

( , , ) ( ) e ^{r x} ( ) 1 e ^{r x}

I x_ν p z ≈I p ^{−τ μ Φ} + S r − ^{−τ μ Φ}

( )

S( ,0 0) S( ,0 0)

behind core 0

core be

CMF

CMF

fore

This means that the resonance zone (where

( , , ) ( ) e ( ) 1 e = const

whilst the resonance zone (where

behind ( ) 1)

befo

( )

re ( ) 0

(

)

, , )

r r

I x p z I p S r

I p

I x p z

x

x

τ μ τ μ

ν

ν

− −

≈ + −

Φ

≈

=

Φ =

*

Only inside the resonance zone, the optical depth increases, and the intensity v

= const for 0

aries accordingly!

else

To calculate the in Sobole

com

v a

pare wit

inten

h

sity pp

p R

⎫⎪

⎪⎪

⎪⎪⎬

≤ ⎪

⎧ ⎪

⎨ ⎪

⎩ ⎪

⎪⎭

roximation ( ), the

location of the resonance zone has to be evaluated f

requir or eac

ed, e.g., for the emergent pro h frequency and impact parame

file ter!

slides 23 / 24

USM

Sobolev theory

( )

S( ,0 0) ( CMF) S( ,0 0) ( CMF)

core 0 CMF

CMF

Now comes the 2nd "trick" .... As outlined, we first calculate

( , ) ( ) e ( ) 1 e ( ( , )) d

Again, we find a contribution only for 0, i.

r x r x

I r I p S r x r x

x

τ μ τ μ

μ

φ μ

−∞

⎡ ⎤

= ⎣ + − ⎦

≈

∫

e., v'( ). x ≈ μ r

S CMF S

0 0

CMF

CMF CMF

( , ) ( ) ( ,

core

Thus, we can replace by and by ; realizing that ( ) d d

with ( v') 1 for

and ( v') 0 for , we find

( , ) ( ) e

( ) 1 e

r r

x x

x x x

x x x

I r I p

S r

μ μ

φ

μ μ

μ

= − Φ

Φ = − → → −∞

Φ = − → → +∞

≈ + ( −

)

S S

1

) ( )

0

( , ) ( , )

core

S S

d

1 e 1 e

( ) ( ) 1 ,

( , ) ( , )

which is already PURELY LOCAL .

x

r r

I p S r

r r

τ μ τ μ

τ μ τ μ

Φ

− −

⎡ ⎤ Φ =

⎣ ⎦

⎛ ⎞

− −

= + ⎜ − ⎟

⎝ ⎠

∫

USM

Sobolev theory

XXIX Canary Island

( )

S( , ) S( , )

core

S S

c core

c co

1 e 1 e

( , ) ( ) ( ) 1

( , ) ( , )

Finally, by integrating over d , and accounting for the limits regarding the first term,

( ) ( ) 1 ( ) ( ) with

( )

r r

I r I p S r

r r

J r r I r S r

r I

τ μ τ μ

μ τ μ τ μ

μ

β β

β

−

⎛

⎞

− −

= + ⎜ − ⎟

⎝ ⎠

= + −

S S

*

1 ( , ) 1 ( , )

re core

S 1 S

1 1 e 1 1 e

( , ) d and escape probability ( ) d .

2 ( , ) 2 ( , )

r r

I r

r r

τ μ τ μ

μ

μ ν μ β μ

τ μ τ μ

− −

−

− −

= ∫ = ∫

(ii) the angular integration does NOT require a highly resolved angular grid (since the interaction between , , and has been already accounted for) x μ r

( )

that

(i) the core intensity has to be emitted (evaluated) at the core with an observer's (rest) frame frequency of 1 v( ) / (corresponding to 0, i.e., v') ,

N

in order to d ot

p e

is

r c x x

ν ν ≈ + μ = = μ

CMF

This ensures that the resonance zone is il

lay a CMF-frequency of at ( , v

luminated by the full core-intensity ( )).

(desh

adowing!)

ν ≈ ν μ r